Problem

Let

$$f_n(x)=\sum_{k=2}^n k(k-1)2^{-k}x^{k-2}+\sum_{k=0}^n \frac{x^{k+1}}{k!}$$

for $x\in(-\infty,\infty)$. Which of the following statements is true?

(a) $\lim_{n\to\infty}f_n(x)$ exists for every $x\in(-\infty,\infty)$.

(b) $\lim_{n\to\infty}f_n(x)$ does not exist whenever $|x|>0$.

(c) $\lim_{n\to\infty}f_n(x)$ does not exist whenever $|x|>0.5$.

(d) $\lim_{n\to\infty}f_n(x)$ does not exist whenever $|x|>1$.

(e) None of the above statements is true.